despite what our senses tell us, zeno is telling us that achilles will never catch the tortoise. zeno's paradoxes were a big problem for the greek philosophers, and they did just about everything they could to avoid confronting the infinite because they based their arithmetic and their entire worldview on something much more tangible: geometry. their notion of the mathematical and the physical was intimately linked to the practice of measuring objects using arbitrary but finite units, like the length of a finger or the width of a palm. and as we do today, units like inches or centimeters, these are arbitrary but commonly held divisions of length. now, the greeks believed that given any two lengths, an arbitrary unit of some kind could always be found to measure both lengths in whole-number multiples, meaning that the two lengths are always commensurate, or commensurable. now, pythagoras was perhaps the first to articulate this belief based on whole numbers, and it came from his observations about music. pythagoras noted that if two commensurate strings were strummed to vibrate, t